Knot at Lunch, July 5, 2007

Knot at Lunch/Navigation  Date(s) Link(s) 2010/02/22 ??? 2010/01/20 Formal integration 2010/01/13 Combing wB 2010/01/06 Exponentiation in tder 2009/09/22 descending v-knots 2009/08/26 red over green v-tangles 2009/08/19 Polyak Algebra 2009/07/08 Immanants 2009/07/01 Alexander modules 2009/06/24 Alexander modules 2009/06/10 Alexander, PBW for A^w, class videos 2009/06/03 Low key 2009/05/06 Low key 2009/04/29 Winter on Ribbons 2009/04/22 Misc 2009/04/15 KV 2009/03/25 KV 2009/03/18 Peter Lee 2009/03/04 Kirby calculus 2009/02/25 Karene on Reidemeister-Schreier 2009/02/11 Dror on Trotter, Jana on Alexander 2009/02/04 Bracelets 2009/01/28 gl(N) chickens 2009/01/15 2D Gauss Diagrams, FiC 2009/01/08 S&G update and more 2008/12/11 Chu on Garside, II 2008/12/04 wZ is 1-1 2008/11/27 The Wen 2008/11/20 The Zoom Space 2008/11/13 Chu on Garside 2008/11/06 Z and GPV 2008/10/30 Peter Lee on EHKR 2008/10/23 Map of the Field 2008/09/25 Hirasawa on Open Books 2008/09/18 Odd Khovanov 2008/09/17 Categorification.m 2008/09/11 More wAlex 2008/09/03 ? 2008/08/27 Dexp and BCH 2008/08/06 Z, A, det, tr, log 2008/07/30 Alexander Relations Marathon 2008/07/02 Peter Lee on horizontal Aw 2008/06/25 w-Alexander 2008/06 16-22 Thomas Fiedler Marathon 2008/06/11 ? 2008/06/04 Dylan Thurston 2008/05/28 Welded Tangles 2008/05/21 Bruce, Lucy 2008/04/23 Welded Knots 2008/04/16 Quandles and Lie algebras 2008/04/09 No-div Alekseev-Torossian 2008/04/02 Knotted Kung Fu Pandas 2008/03/26 Homotopy invariants 2008/03/19 Infinitesimal Artin 2008/03/12 Infinitesimalization of Artin 2008/03/05 Krzysztof Putyra on Odd Khovanov Homology 2008/02/27 Karene Chu on Proof of Artin 2008/02/20 Organizational, Hecke algebras 2008/02/13 Exponential and Magnus expansions 2008/02/06 cancelled 2008/01/30 Artin's theorem 2008/01/16 Hutchings' work, 2 2008/01/09 Hutchings' work, 1 2007/12/12 Bone soup 2007/12/05 Expansions 2007/11/28 Quantum groups 2007/11/21 Surfaces and gl(N)/so(N) 2007/11/07 Expansions for Groups 2007/10/31 Louis Leung on bialgebra weight systems 2007/10/24 Zsuzsi Dancso, continued 2007/10/17 Jana Archibald on the multivariable Alexander 2007/10/10 Zsuzsi Dancso on diagrammatic su(2) 2007/10/03 Hernando Burgos on alternating tangles 2007/09/26 Peter Lee on homology 2007/09/06 Garoufalidis' visit 2007/08/30 Art and enumeration 2007/08/23 My Hanoi talk? 2007/08/16 Lie bialgebra weight systems and more 2007/07/19 Subdiagram formulas 2007/07/12 Playing with Brunnians 2007/07/05 Virtualization 2007/06/28 Virtual braids 2007/06/07 Virtual knots 2007/05/31 Social gathering 2007/05/24 Lee on Frozen Feet

Invitation

Dear Knot at Lunch People,

We will have our next summer lunch on Thursday July 5, 2007, at the usual place, Bahen 6180, at 12 noon.

As always, please bring brown-bag lunch and fresh ideas. I'm not sure what we will be talking about; perhaps just continue with last week's topics.

As always, if you know anyone I should add to this mailing list or if you wish to be removed from this mailing list please let me know. To prevent junk accumulation in mailboxes, I will actively remove inactive people unless they request otherwise.

Best,

Dror.

Some Content

Definition. Let $\varphi :B\to S$ be a group homomorphism; denote its action by $b\mapsto {\bar {b}}$; i.e., let ${\bar {b}}:=\varphi (b)$ for every $b\in B$. Let "the virtualization $VB$ of $B$", or more precisely, "the virtualization $VB_{\varphi }$ of $B$ with respect to $\varphi$", be the following quotient of the free product $B\star S$ of $B$ and $S$:

$VB:=B\star S\left/s^{-1}b_{1}s=b_{2}\right.$ whenever $s\in S$, $b_{1,2}\in B$ and $s^{-1}{\bar {b}}_{1}s={\bar {b}}_{2}$ in $S$.

Though note that we do not mod out by $b^{-1}s_{1}b=s_{2}$ when $s_{1,2}\in S$, $b\in B$ and ${\bar {b}}^{-1}s_{1}{\bar {b}}=s_{2}$.

It is clear that $\varphi$ extends to a homomorphism ${\hat {\varphi }}:VB\to S$. Let "the pure virtualization $PVB$ of $B$" be the kernel of that homomorphism:

$PVB:=\ker {\hat {\varphi }}\subset VB$.

Question. Is this definition at all interesting? More precisely:

• If $B$ is a braid group and $S$ is the corresponding symmetric group, can $VB$ be reasonably identified with "virtual braids"?
• Does the $PVB$ that we get here agree with $PVB_{n}$ of last time?
• Is this definition encountered anywhere else in mathematics?
• Are there other examples in which this definition is interesting?
• Do we gain any new insight by using this definition?